Given a right triangle where side $\mathrm{AB}$ is $5$ units, side $\mathrm{BC}$ is $12$ units, and $\mathrm{AC}$ is the hypotenuse, which is the best approximation for the measure of angle $\mathrm{ABC}$?

• A. $30$ degrees
• B. $45$ degrees
• C. $22.6$ degrees
• D. $67.4$ degrees

Answer: (C)

Verified step by step guidance

1. Identify the sides of the right triangle relative to angle ABC. Here, side AB is adjacent to angle ABC, side BC is opposite to angle ABC, and AC is the hypotenuse.
2. Recall that the tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side. So, \( \tan(\angle ABC) = \frac{\text{opposite}}{\text{adjacent}} = \frac{BC}{AB} \).
3. Substitute the given side lengths into the tangent ratio: \( \tan(\angle ABC) = \frac{12}{5} \).
4. To find the measure of angle ABC, take the inverse tangent (arctangent) of the ratio: \( \angle ABC = \tan^{-1}\left(\frac{12}{5}\right) \).
5. Use a calculator to evaluate \( \tan^{-1}\left(\frac{12}{5}\right) \) to find the angle measure in degrees, which will give the best approximation for angle ABC.